Pressure Derivative on Uncountable Alphabet Setting: a Ruelle Operator Approach

TL;DR

This paper applies a Ruelle operator approach to compute pressure derivatives for translation invariant spin systems on uncountable alphabets, proving absence of metastable states and differentiability of pressure in certain models.

Contribution

It introduces a Ruelle operator framework for uncountable alphabet spin systems, establishing pressure derivative formulas and differentiability results.

Findings

Proves absence of metastable states for continuous potentials.

Shows Fréchet differentiability of pressure in a Heisenberg-type model.

Establishes exponential decay of two-point functions at positive temperature.

Abstract

In this paper we use a recent version of the Ruelle-Perron-Frobenius Theorem to compute, in terms of the maximal eigendata of the Ruelle operator, the pressure derivative of translation invariant spin systems taking values on a general compact metric space. On this setting the absence of metastable states for continuous potentials on one-dimensional one-sided lattice is proved. We apply our results, to show that the pressure of an essentially one-dimensional Heisenberg-type model, on the lattice $\mathbb{N}\times \mathbb{Z}$, is Fr\'echet differentiable, on a suitable Banach space. Additionally, exponential decay of the two-point function, for this model, is obtained for any positive temperature.